$\phi$-$\delta$-$S$-primary hyperideals

Among many generalizations of primary hyperideals, weakly $n$-ary primary hyperideals and $n$-ary $S$-primary hyperideals have been studied recently. Let $S$ be an $n$-ary multiplicative set of a commutative Krasner $(m,n)$-hyperring $K$ and, $\phi$ and $\delta$ be reduction and expansion functions of hyperideals of $K$, respectively. The purpose of this paper is to introduce $n$-ary $\phi$-$\delta$-$S$-primary hyperideals which serve as an extension of $S$-primary hyperideals with the help of $\phi$ and $\delta$. We present some main results and examples explaining the sructure of this concept. We examine the relations of $n$-ary $S$-primary hyperideals with other classes of hyperideals and give some ways to connect them. Moreover, we give some characterizations of this notion on direct product of commutative Krasner $(m, n)$-hyperrings.